Passengers on board a stricken US cruise liner have spoken of dwindling food supplies and malfunctioning toilets after a fire knocked out the ship's engines, electricity and air conditioning over the weekend.

The Carnival Triumph lost power after a fire in the aft engine room early Sunday. The ship, which is carrying 3,143 passengers and 1,086 crew members, drifted for about 90 miles in the Gulf of Mexico before tugboats intercepted it on Monday. It is expected to dock in Mobile, Alabama, on Thursday.

The lack of electricity has meant overflowing toilets and soaring temperatures on board, with many passengers sleeping on deck. Without power, the Carnival Triumph's stabilisers were not functioning on Monday, and passengers complained that the ship's subsequent listing from side to side was doing little to improve the troublesome bathroom facilities.

Joel Dyer, a youth worker at a church in Oklahoma City, told the Guardian that his wife Joy, 36, is on the cruise ship with a group of friends, and is among a number of those who have taken to the deck to escape the heat in their cabins. In text messages sent from the ship on Sunday and Monday and shared with the Guardian, Dyer said conditions were deteriorating.

"Ship has been leaning all day, but night winds are made it lean more – it was very tilted to the side, hard to walk across," she said. "Getting very resourceful– making more camp setups with bath robe belts, bent hangers, sheets, room tables, unwound rope that we take from other parts of the ship."

On Sunday, passengers were forced to use showers and plastic bags instead of toilets, Dyer and several other passengers have confirmed. By Monday a few toilets were back in use but there were long lines.

"People are being good – crew is so unselfish and working so hard," Dyer said in a text sent on Monday. "People starting to get more irritable, and others showing more kindness. I fear a couple of days from now when people start going crazy."

Joel, 37, who is at home in Yukon, Oklahoma with the couple's two sons, said "the most difficult part is not hearing much directly" from his wife. "Apart from the times when they were able to get cellular reception from the ships that were getting supplies, those were the only times I get to hear from her."

Brytan Thomas, from Texas, told the Guardian a similar story of brief text exchanges with her mother and sister, who are onboard the Carnival Triumph.

"The conditions are terrible," Thomas said. "Water is seeping from the walls, people are sleeping on the deck outside, they have very limited food and stuff to drink and not much power. The boat is leaning and when people walk [they] walk sideways. It is very stressful and I am just praying that they make it to Alabama safely Thursday."

The Carnival Triumph set sail from Galveston, Texas, on Thursday 7 February, and was due to return to the port on Monday morning. The 272-metre cruise liner ran into difficulty on Sunday morning when a fire broke out in the engine room. The fire was automatically extinguished, with no injuries to crew or passengers, but it left the ship without power, drifting off the Yucatan Peninsula in the Gulf of Mexico.

In a statement, Carnival Cruise Lines president Gerry Cahill said the ship would arrive in Mobile, Alabama, "sometime on Thursday".

"All of our guests are safe, and we're doing everything we can to make them as comfortable as possible. The ship has maintained emergency generator power since the fire occurred and the technical team on board has been successful in gradually restoring auxiliary power to operate some basic hotel functions," he said in a statement.

"Currently, public and cabin toilets are operational in certain sections of the ship, power has been restored to a limited number of elevators, and some power in the Lido dining area is providing for hot coffee and limited hot food service."

Cahill said all passengers would receive a full refund for the cruise and transport expenses. Carnival would also offer a "future cruise credit equal to the amount paid for this voyage, as well as reimbursement of all shipboard purchases during the voyage," he said.

The Carnival Triumph and the company that owns it have faced controversy before. The Triumph was seized in Texas in March 2012 after a lawsuit was filed against Carnival Cruise Lines on behalf of a woman who died on the Costa Concordia cruise ship in Italy.